We explore the pulsar slot gap (SG) electrodynamics up to very high altitudes, where for most relatively rapidly rotating pulsars both the standard small-angle approximation and the assumption that the magnetic field lines are ideal stream lines break down. We address the importance of the electrodynamic conditions at the SG boundaries and the occurrence of a steady-state drift of charged particles across the SG field lines at very high altitudes. These boundary conditions and the cross-field particle motion determine the asymptotic behavior of the scalar potential at all radii from the polar cap (PC) to near the light cylinder. As a result, we demonstrate that the steady-state accelerating electric field, E(sub ll), must approach a small and constant value at high altitude above the PC. This E(sub ll) is capable of maintaining electrons moving with high Lorentz factors (approx. a few x 10(exp 7)) and emitting curvature gamma-ray photons up to nearly the light cylinder. By numerical simulations, we show that primary electrons accelerating from the PC surface to high altitude in the SG along the outer edge of the open field region will form caustic emission patterns on the trailing dipole field lines. Acceleration and emission in such an extended SG may form the physical basis of a model that can successfully reproduce some pulsar high-energy light curves.